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# Randomized Blocks Designs: Omnibus vs. Pairwise Comparison, Fixed vs. Relative Optimal Discriminant Threshold, and Raw vs. Ipsative z-Score Measures

Paul R. Yarnold & Ariel Linden

Optimal Data Analysis, LLC & Linden Consulting Group, LLC

This study extends recent research assessing the use of relative thresholds in matched-pairs designs, for a randomized blocks design in which four treatments are randomly assigned to blood samples drawn from each of eight people (each person treated as a block). Both raw and ipsatively standardized plasma clotting times are compared between treatments.

# Using Fixed and Relative Optimal Discriminant Thresholds in Randomized Blocks (Matched-Pairs) Designs

Paul R. Yarnold & Ariel Linden

Optimal Data Analysis, LLC & Linden Consulting Group, LLC

Optimal discriminant analysis (ODA) is often used to compare values of one (or more) attributes between two (or more) groups of observations with respect to a fixed discriminant threshold that maximizes accuracy normed against chance for the sample. However, a recent study using a matched-pairs design found that using a relative discriminant threshold to assess an (exploratory or confirmatory) *a priori* hypothesis separately for each pair of observations can identify inter-group differences which otherwise are too subtle to be identified by using fixed thresholds. The present investigation replicates the finding regarding efficacy of relative thresholds for matched-pairs designs, this time for a randomized blocks design consisting of two patient groups (one group assigned to take an antidepressant drug, the other group assigned to take a placebo) between which a numerical measure of depression was compared. Several recommendations are made concerning use of improved modern optimal statistical alternatives for this class of experimental design.

# ODA vs. t-Test: Lysozyme Levels in the Gastric Juice of Patients with Peptic Ulcer vs. Normal Controls

Paul R. Yarnold

Optimal Data Analysis, LLC

Lysozyme levels in gastric juice of peptic ulcer patients were compared against normal controls by *t*-test, finding *p*<0.05. Because standard deviations differed by a factor of two between groups, and were proportional to the means, analysis of natural logarithms was instead deemed appropriate: the resulting *t*-test wasn’t statistically significant. Analyzed by ODA no statistically significant between-group difference emerged, and results obtained for raw data and for natural logarithms were identical because ODA results (i.e., *p* and ESS) are invariant over all monotonic transformations of the data.

# Regression vs. Novometric-Based Assessment of Inter-Examiner Reliability

Paul R. Yarnold

Optimal Data Analysis, LLC

Four examiners independently recorded the DMFS (decayed, missing, filled surfaces) scores of ten patients. Inter-examiner correspondence of DMFS scores was evaluated using Pearson correlation and novometric analysis. Whereas essentially perfect correlation models were unable to accurately predict DMFS scores in training analysis, novometric models were consistently perfect in both training and reproducibility analysis.

# Fixed vs. Relative Optimal Discriminant Thresholds: Pairwise Comparisons of Raters’ Ratings for a Sample

Paul R. Yarnold

Optimal Data Analysis, LLC

Foundational to the ODA algorithm when used with an ordered attribute is the identification of the optimal threshold—the specific cutpoint that yields the most accurate (weighted) classification solution for a sample of observations. ODA models involving a single optimal threshold will henceforth be called “fixed-threshold” models. This note proposes a new “relative-threshold” ODA model for an inter-examiner reliability study in which four examiners independently rate teeth condition for a sample of ten patients: “An important inferential question is whether the rater effects differ significantly from one another” (p. 19). In the original study, analysis of variance showed rater C assigned the greatest mean rating across patients: “The inference is therefore drawn that differential measurement bias exists (i.e., the *k* examiners differ systematically from one another in their mean levels of measurement)” (pp. 20-21). ODA was used to compare the entire response distribution (not only means) between raters. A fixed-threshold model identified no effects. A relative-threshold model tested the hypothesis that, for each observation in the sample considered separately, the rating by rater X will be less than (or equal to) the rating by rater Y. Analysis showed that the distribution of ratings made by rater C was nearly perfectly greater than corresponding (non-discriminable) ratings made by raters A, B, and D. This finding hints of possible development of optimal analogues of multidimensional scaling and facet theory methodologies.

# Logistic Discriminant Analysis and Structural Equation Modeling Both Identify Effects in Random Data

Ariel Linden, Fred B. Bryant & Paul R. Yarndol

Linden Consulting Group, LLC, Loyola University Chicago & Optimal Data Analysis, LLC

Recent research compared the ability of various classification algorithms [logistic regression (LR), random forests (RF), support vector machines (SVM), boosted regression (BR), multi-layer perceptron neural net model (MLP), and classification tree analysis (CTA)] to correctly fail to identify a relationship between a binary class (dependent) variable and ten randomly generated attributes (covariates): only CTA failed to find a model. We use the same ten-variable N=1,000 dataset to assess training classification accuracy of models developed by logistic discriminant analysis (LDA), generalized structural equation modelling (GSEM), and robust diagonally-weighted least-squares (DWLS) SEM for binary outcomes. Except for CTA, all machine-learning algorithms assessed thus far have identified training effects in random data.